Who should engage first?

This seems to be pretty much the same basic development for everyone at my level no matter whether you're white or black. Slight variations occur but I'm not sure they matter much.

At some point, someone will need to engage and push a pawn to the middle to start the exchanges. Is it more beneficial to go first, get taken and react...or wait for the opponent to initiate the real battle? Sometimes when we end up here I don't know if I should initiate or make a random side pawn move that will happen eventually anyway.

Hi. Pushing pawns opens up diagonals and exchanging pawns opens up files. So when you push the pawns in the centre, you may open up the game. As a general rule, when you are ahead in development, with more pieces off the back rank and your king castled, compared with your opponent, opening the centre with pawn exchanges is to your advantage, as you can use the open lines to attack. If you are behind in development, then try to keep the centre closed while you catch up. Definitely don't open the centre while your king is still there, get castled first.

Thanks, I meant to find one where we both had just castled...basically where we end our development an almost exact mirror of each other...who should enter the center first, that first move that really gets things going. Usually both sides engage the back rank bishop one square so they can castle and the other copies.

But thanks on the pushing vs exchanging idea!

Sorry. I didnt realise you meant the exact position shown. I was writing very generally. I cant see the entire board on my phone.

Roger that. As long as you can see the center 16, you should get the idea. Basically we both developed the same and now someone has to initiate what's going to become the first series of trades to establish somethhing

You don't want to approach the game as just exchanging pieces until someone blunders their queen and loses the game. You should be looking to maximize the power of your pieces. Put your rooks on open files, put your knights onto outpost squares in your opponents territory, put your bishops and squares where they control diagonals that aim towards their kingside. Put your queen where it is creating multiple threats. etc. In that position you put in there, it is generally considered to be a slow maneuvering game. in that position i would be looking at nd7 and looking for the c5 pawn break

https://www.chess.com/blog/KeSetoKaiba/opening-principles-again 

That hyperlink should be of use to you

We never really know what will engage in battle first, but usually it is the minor pieces (Knights and Bishops) or pawns during the opening. A little later in the game, we typically see the major pieces (Queen and Rooks) coming into the attack, but every game is different. This pattern of clashing in increasing order of piece value is more of a trend than some rule or guideline to follow; we must adapt to whatever the position calls for!

Chess opening principles is nice because it avoids many traps and opening troubles while usually getting a solid middlegame where we can begin to create plans, relocate forces, attack targets and so on

Oh, was that an episode from one of Aman's Building Habits series?  If so, consider checking the rating he was pretending to be and what rules he was following at the time.  

Opening principles is a very helpful guide,  I'll be checking out the link too.  If you'd like to practice those principles in an unrated game, feel free to send a challenge

Morphy taught us that the fight begins on the first move. If you are playing so passively that you think 'engagement' happens later... you're playing too passively. Maybe too automatically.

OK, maybe that opening you're using is a snoozefest. That might also be part of your problem.

Yeah, in general it's better if you're the one recapturing and your opponent is the one initiating captures.

Rooks on an open file are an easy example... whoever captures first loses control of the file.

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An even easier example is in the opening when the person recapturing gets to develop.

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A more advanced example involves the fact that pieces don't like to be blocked by friendly pieces.

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So (generally) whenever you capture, you want it to do something good for you (or at least be neutral)

Just a few pointers, because it's not about when to make that first big engaging move, you should already know how you're going to handle that before the game begins.  The key is in understanding THE single most basic principle, and just a few main-line openings.

Big secret reveal time:

Chess is a game where WHITE ATTACKS! and Black defends.  Ergo:

1) When the game starts, are you black?  Then get into the defensive mindset.

2a) On WHITE, PLAY AGGRESSIVELY to diminish your opponent's tempo.  TEMPO is an actual chess word, it means how your game is flowing, how your attack is flowing.  ATTACK, ATTACK, ATTACK!!!

2b) All of the WHITE initiated main-line chess openings are built around the above (2a) principle.  The mainline systems WHITE plays are built around possibly achieving D4! at some point, if the opening is 1)E4, (eg: Ruy Lopez, Italian, Scotch, Petrof, those boring systems lower level players usually gravitate towards.  Well, Magnus & Caruana have been playing The Berlin variant of Ruy, so, there are exceptions...) Anyway...

2c) For Chis Sake, on White don't squander your INITIATIVE for attacking, maintain your TEMPO, and trounce Black.  If Black lets you build a solid D4 & E4! center in the opening, take it and never look back.

2d) At times, your Black opponent will play a soft response to 1)E4, like The French, or RETI.  Take that opportunity to launch your queen pawn straight out to 2)D4!  Those "soft" Black openings are only for advanced players who are trying to lure White into the center, so Black can launch a preplanned attack of White's over extended center.  If your opponent does not realize this, you have a tremendous opportunity to push your TEMPO down the opponent's throat.  Only if you suspect that your Black opponent actually understands those more sublime systems would you hold back your pieces and play more conservatively.

3) On Black, I generally play Sicilian.  It's a reliable "go-to" system for Black to maybe take control of the game in which White usually opens 1)E4 at your level of play.  Also, you'll know right away if White understands Sicilian because they'll probably respind with something like the Alapin.  For me on Black, I will transition into the Dragon or the Najdorf, depending on White's role out move of the King bishop.  If White plays an Italian King bishop to C5, opt for the Dragon, and be ready to land your Queen Knight on D4 to stop a late development fried liver.

The general ideas about capturing I mentioned above can't really be used to talk about the position you give though... now I'll talk about your position, except I'll make it even more annoying for new players

Let's say both sides did everything you're "supposed" to do. They moved all their knights and bishops off the back rank, they castled, they also moved their queen off the back rank (but not too far from home), they even centralized the rooks.

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When development is complete, generally attention turns to pawn breaks and infiltration.  You can play pawn breaks and infiltration earlier, but when development is complete there's not much else to do.

White would like to play the pawn break c2-c4 but the knight is blocking the pawn. White can move to black's side of the board with Ne4, which also frees his bishop to go to f3, so I think that's good.

A more advanced answer is while both sides have developed, neither side has developed very well. It's useful to point out that in the position you give both players have blocked their c pawns with their knights. In positions with pawns on d4 and d5 the c pawn very often plays an important role. It moves one square forward to support the central d pawn, or it moves two squares forward to attack the opponent's center pawn.

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The same doesn't happen for positions with pawns on e4 and e5 because moving the f pawn exposes the king.

Anyway, the main point I want to make for you is to move a game forward look for pawn breaks and moving to the opponent's side. The more advanced point is that when a position is very awkward sometimes looking to see how you could have avoided it in the first place is useful. In the position I gave both sides made many sub-par development decisions... going over all of them probably wont be so useful for you, but sometimes when asking for advice, you might try "could I have developed differently to avoid this problem?"

Yes!, that's the exact video...so you can see where I'm coming from. Can't play anymore tonight, but I'll friend you and we can play later.

Yes, these examples are exactly what I was looking for. There's a clear advantage to initiating and recapturing depending on the circumstance. Your simitricle example is exactly the position I was talking about and the knight IS usually the option I go for. Sometimes, I judge the opponent and make "less than perfect" moves because It appears I can bait them into making costly mistakes.  Thank you guys a bunch.

"Sometimes when we end up here I don't know if I should initiate or make a random side pawn move that will happen eventually anyway."
Make neither random moves nor side pawn moves. Every move should have a purpose. Moving side pawns is often loss of tempo and weakening. In the given position knights and queen bishops are out, but king's bishops not yet. The kings are not yet castled, the kings are still in the center and the rooks are not yet connected. The logical move is Bd3 bringing the king's bishop out and preparing O-O. Only when all pieces are into play can you start regrouping to improve your worst placed piece. As long as there are pieces on their starting squares, they are the worst placed pieces.

Best move in your example for black should be knight to B4 threating a fork on c2.

John Bartholomew's Chess Fundamentals Series on you tube has a video on trades which might help. Also one on pawn play which goes into timing of pawn breaks. There may be a number of ways to play the position, for example you could chose to advance pawns on the queenside to gain space and find better stable squares for your pieces, leaving the centre as it is. A reason for encouraging your opponent to capture first is if the recapture favours you, so you want to occupy the square rather than remove the piece (or both). With rooks, this might mean winning control of the file. When trading equal material, its not necessarily the pieces being taken off the board that matter, rather the resulting position afterwards, or both.